Estimation and Prediction of Functional Autoregressive Processes

We present a generalization of some previous works (Bosq, Mourid, Pumo) about the functional forecast of a Banach autoregressive processes. We are mainly concerned with order p , p >1, autoregressive processes which appear to be a natural extension of the well-known R d -valued autoregressive processes to a functional framework. This modelization provides an new approach for estimating and for predicting a continuous time stochastic process over an entire time interval. Using results from [12] we prove asymptotic properties of estimators of the parameters and predictors which are based upon a principal component decomposition of a Hilbert-Schmidt operator with unknown eigenvectors.